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Properties

Label 3312.2.a.o.1.1
Level $3312$
Weight $2$
Character 3312.1
Self dual yes
Analytic conductor $26.446$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3312,2,Mod(1,3312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3312.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.4464531494\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3312.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{5} +4.00000 q^{7} +O(q^{10})\) \(q+2.00000 q^{5} +4.00000 q^{7} -2.00000 q^{11} +7.00000 q^{13} +4.00000 q^{17} +6.00000 q^{19} -1.00000 q^{23} -1.00000 q^{25} -5.00000 q^{29} -3.00000 q^{31} +8.00000 q^{35} +2.00000 q^{37} +9.00000 q^{41} -8.00000 q^{43} -1.00000 q^{47} +9.00000 q^{49} +6.00000 q^{53} -4.00000 q^{55} -8.00000 q^{59} -10.0000 q^{61} +14.0000 q^{65} -2.00000 q^{67} -13.0000 q^{71} -3.00000 q^{73} -8.00000 q^{77} -6.00000 q^{79} +8.00000 q^{85} +4.00000 q^{89} +28.0000 q^{91} +12.0000 q^{95} -8.00000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 7.00000 1.94145 0.970725 − 0.240192i \(-0.0772105\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 − 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 −0.145865 −0.0729325 − 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.0000 1.73649
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.0000 −1.54282 −0.771408 − 0.636341i \(-0.780447\pi\)
−0.771408 + 0.636341i \(0.780447\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 − 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 28.0000 2.93520
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 − 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 − 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 − 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) 24.0000 2.08106
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.0000 −1.70872 −0.854358 − 0.519685i \(-0.826049\pi\)
−0.854358 + 0.519685i \(0.826049\pi\)
\(138\) 0 0
\(139\) 19.0000 1.61156 0.805779 − 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −14.0000 −1.17074
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −11.0000 −0.895167 −0.447584 − 0.894242i \(-0.647715\pi\)
−0.447584 + 0.894242i \(0.647715\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 − 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 36.0000 2.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.00000 0.373718 0.186859 − 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 − 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 − 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) 1.00000 0.0719816 0.0359908 − 0.999352i \(-0.488541\pi\)
0.0359908 + 0.999352i \(0.488541\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000 0.213741 0.106871 − 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 − 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −20.0000 −1.40372
\(204\) 0 0
\(205\) 18.0000 1.25717
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 28.0000 1.88348
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.00000 0.530979 0.265489 − 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 − 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 −0.196537 −0.0982683 − 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 0 0
\(235\) −2.00000 −0.130466
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.0000 1.22901 0.614504 − 0.788914i \(-0.289356\pi\)
0.614504 + 0.788914i \(0.289356\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) 42.0000 2.67240
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.0000 −0.935674 −0.467837 − 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.0000 1.60323 0.801614 − 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.0000 0.914566 0.457283 − 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 17.0000 1.02143 0.510716 − 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 − 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 22.0000 1.30776 0.653882 − 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 36.0000 2.12501
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −16.0000 −0.934730 −0.467365 − 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.931556
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.00000 −0.404820
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 − 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) −7.00000 −0.388290
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 − 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −28.0000 −1.52526 −0.762629 − 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 25.0000 1.33822 0.669110 − 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −33.0000 −1.75641 −0.878206 − 0.478282i \(-0.841260\pi\)
−0.878206 + 0.478282i \(0.841260\pi\)
\(354\) 0 0
\(355\) −26.0000 −1.37994
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 − 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 − 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −35.0000 −1.80259
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 36.0000 1.83951 0.919757 − 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.00000 0.405616 0.202808 − 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −1.00000 −0.0501886 −0.0250943 − 0.999685i \(-0.507989\pi\)
−0.0250943 + 0.999685i \(0.507989\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) −21.0000 −1.04608
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 − 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −32.0000 −1.57462
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.00000 0.390826 0.195413 − 0.980721i \(-0.437395\pi\)
0.195413 + 0.980721i \(0.437395\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 − 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 − 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 −0.287019
\(438\) 0 0
\(439\) −13.0000 −0.620456 −0.310228 − 0.950662i \(-0.600405\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −25.0000 −1.18779 −0.593893 − 0.804544i \(-0.702410\pi\)
−0.593893 + 0.804544i \(0.702410\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 − 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 56.0000 2.62532
\(456\) 0 0
\(457\) −32.0000 −1.49690 −0.748448 − 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.00000 −0.0465746 −0.0232873 − 0.999729i \(-0.507413\pi\)
−0.0232873 + 0.999729i \(0.507413\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.00000 0.0925490 0.0462745 − 0.998929i \(-0.485265\pi\)
0.0462745 + 0.998929i \(0.485265\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16.0000 −0.726523
\(486\) 0 0
\(487\) −1.00000 −0.0453143 −0.0226572 − 0.999743i \(-0.507213\pi\)
−0.0226572 + 0.999743i \(0.507213\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 29.0000 1.30875 0.654376 − 0.756169i \(-0.272931\pi\)
0.654376 + 0.756169i \(0.272931\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −52.0000 −2.33252
\(498\) 0 0
\(499\) 25.0000 1.11915 0.559577 − 0.828778i \(-0.310964\pi\)
0.559577 + 0.828778i \(0.310964\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −26.0000 −1.15928 −0.579641 − 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(504\) 0 0
\(505\) 28.0000 1.24598
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −32.0000 −1.41009
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 − 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) −24.0000 −1.04945 −0.524723 − 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 63.0000 2.72883
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −3.00000 −0.128980 −0.0644900 − 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 39.0000 1.66752 0.833760 − 0.552127i \(-0.186184\pi\)
0.833760 + 0.552127i \(0.186184\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −30.0000 −1.27804
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 − 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) −56.0000 −2.36855
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 24.0000 1.00969
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 − 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 − 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −39.0000 −1.62359 −0.811796 − 0.583942i \(-0.801510\pi\)
−0.811796 + 0.583942i \(0.801510\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.0000 −0.701665 −0.350833 − 0.936438i \(-0.614101\pi\)
−0.350833 + 0.936438i \(0.614101\pi\)
\(588\) 0 0
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 32.0000 1.31187
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 − 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 − 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 4.00000 0.162355 0.0811775 − 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.00000 −0.283190
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 − 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −40.0000 −1.60774 −0.803868 − 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 − 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 14.0000 0.555573
\(636\) 0 0
\(637\) 63.0000 2.49615
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 − 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 − 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.0000 −1.61188 −0.805938 − 0.592000i \(-0.798339\pi\)
−0.805938 + 0.592000i \(0.798339\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.0000 −0.586995 −0.293498 − 0.955960i \(-0.594819\pi\)
−0.293498 + 0.955960i \(0.594819\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.00000 0.311636 0.155818 − 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 48.0000 1.86136
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 − 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −34.0000 −1.30673 −0.653363 − 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 0 0
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 37.0000 1.41577 0.707883 − 0.706330i \(-0.249650\pi\)
0.707883 + 0.706330i \(0.249650\pi\)
\(684\) 0 0
\(685\) −40.0000 −1.52832
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 42.0000 1.60007
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 − 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 38.0000 1.44142
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12.0000 −0.453234 −0.226617 − 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 56.0000 2.10610
\(708\) 0 0
\(709\) 16.0000 0.600893 0.300446 − 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.00000 0.112351
\(714\) 0 0
\(715\) −28.0000 −1.04714
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −64.0000 −2.38348
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.00000 0.185695
\(726\) 0 0
\(727\) −6.00000 −0.222528 −0.111264 − 0.993791i \(-0.535490\pi\)
−0.111264 + 0.993791i \(0.535490\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 − 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −29.0000 −1.06678 −0.533391 − 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 − 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −22.0000 −0.800662
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 − 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −5.00000 −0.181250 −0.0906249 − 0.995885i \(-0.528886\pi\)
−0.0906249 + 0.995885i \(0.528886\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −56.0000 −2.02204
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −12.0000 −0.431610 −0.215805 − 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) 0 0
\(775\) 3.00000 0.107763
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 54.0000 1.93475
\(780\) 0 0
\(781\) 26.0000 0.930353
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 − 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 48.0000 1.70668
\(792\) 0 0
\(793\) −70.0000 −2.48577
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.0000 0.492214 0.246107 − 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 − 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 5.00000 0.174289 0.0871445 − 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 72.0000 2.47688
\(846\) 0 0
\(847\) −28.0000 −0.962091
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −43.0000 −1.46885 −0.734426 − 0.678689i \(-0.762549\pi\)
−0.734426 + 0.678689i \(0.762549\pi\)
\(858\) 0 0
\(859\) 25.0000 0.852989 0.426494 − 0.904490i \(-0.359748\pi\)
0.426494 + 0.904490i \(0.359748\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −39.0000 −1.32758 −0.663788 − 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) 0 0
\(865\) −28.0000 −0.952029
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12.0000 −0.404290 −0.202145 − 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 39.0000 1.30949 0.654746 − 0.755849i \(-0.272776\pi\)
0.654746 + 0.755849i \(0.272776\pi\)
\(888\) 0 0
\(889\) 28.0000 0.939090
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) 10.0000 0.334263
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 36.0000 1.19668
\(906\) 0 0
\(907\) −14.0000 −0.464862 −0.232431 − 0.972613i \(-0.574668\pi\)
−0.232431 + 0.972613i \(0.574668\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 − 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 − 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −91.0000 −2.99530
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 41.0000 1.34517 0.672583 − 0.740022i \(-0.265185\pi\)
0.672583 + 0.740022i \(0.265185\pi\)
\(930\) 0 0
\(931\) 54.0000 1.76978
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 − 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −9.00000 −0.293080
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −49.0000 −1.59229 −0.796143 − 0.605108i \(-0.793130\pi\)
−0.796143 + 0.605108i \(0.793130\pi\)
\(948\) 0 0
\(949\) −21.0000 −0.681689
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 − 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −80.0000 −2.58333
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −37.0000 −1.18984 −0.594920 − 0.803785i \(-0.702816\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 − 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 0 0
\(973\) 76.0000 2.43645
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 10.0000 0.318950 0.159475 − 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 28.0000 0.887660
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3312.2.a.o.1.1 1
3.2 odd 2 368.2.a.f.1.1 1
4.3 odd 2 1656.2.a.g.1.1 1
12.11 even 2 184.2.a.b.1.1 ✓ 1
15.14 odd 2 9200.2.a.l.1.1 1
24.5 odd 2 1472.2.a.d.1.1 1
24.11 even 2 1472.2.a.k.1.1 1
60.23 odd 4 4600.2.e.f.4049.1 2
60.47 odd 4 4600.2.e.f.4049.2 2
60.59 even 2 4600.2.a.k.1.1 1
69.68 even 2 8464.2.a.m.1.1 1
84.83 odd 2 9016.2.a.j.1.1 1
276.275 odd 2 4232.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.b.1.1 ✓ 1 12.11 even 2
368.2.a.f.1.1 1 3.2 odd 2
1472.2.a.d.1.1 1 24.5 odd 2
1472.2.a.k.1.1 1 24.11 even 2
1656.2.a.g.1.1 1 4.3 odd 2
3312.2.a.o.1.1 1 1.1 even 1 trivial
4232.2.a.e.1.1 1 276.275 odd 2
4600.2.a.k.1.1 1 60.59 even 2
4600.2.e.f.4049.1 2 60.23 odd 4
4600.2.e.f.4049.2 2 60.47 odd 4
8464.2.a.m.1.1 1 69.68 even 2
9016.2.a.j.1.1 1 84.83 odd 2
9200.2.a.l.1.1 1 15.14 odd 2